Test sequences and formal solutions over hyperbolic groups

Given a group G and a sentence $\sigma$ in the language of groups, we say that $\sigma$ is a true sentence in G (denoted by $G\models \sigma$) if $\sigma$ is true if interpreted in G. The first-order or elementary theory of a group G, denoted by Th(G), is the set of all sentences true in G. The groups whose first-order theory we are mainly interested in, are free and hyperbolic groups. The interest of model theorist in the first-order theory of free groups dates back to the 1940s when Alfred Tarski posed his famous questions, which are now widely known as the Tarski-problems, namely if all non-abelian free groups have the same elementary theory and if this theory is decidable. In 2006 Z. Sela and independently O. Kharlampovich and A. Myasnikov showed that $\Th(F_n)=\Th(F_m)$ for all $m,n\geq 2$, answering Tarski's question to the positive. Subsequently Z. Sela generalized his techniques used in his proof of the Tarski problems to classify all finitely generated groups elementary equivalent to a given torsion-free hyperbolic group. One important step in his analysis of the elementary theory of free and torsion-free hyperbolic groups is a generalization of the following theorem of Merzlyakov.\\ Let $F_k=F(a)$ be a non-abelian free group and suppose that $$F_k\models \forall y\ \exists x\ \Sigma(x,y,a)=1\wedge\Psi(x,y,a)\neq 1.$$ Then there exists a retraction $\pi: \langle x,y,a\ |\ \Sigma(x,y,a)=1\rangle \to F(y)\ast F_k$ and $$F_k\models\exists y\ \Psi(\pi(x),y,a)\neq 1.$$ Sela's generalized Merzlyakov's Theorem covers the case that the universal variables y are restricted to some variety. In our work we prove a version of the Generalized Merzlyakov's Theorem over hyperbolic groups (possibly with torsion). We moreover give an example of a Makanin-Razborov diagram for a group with non-trivial JSJ decomposition and conclude that limit groups are in general not freely subgroup separable. We then classify all JSJ decomposition of doubles of free groups of rank two.

Rechte

Nutzung und Vervielfältigung:

Keine Lizenz. Es gelten die Bestimmungen des deutschen Urheberrechts (UrhG).

Bitte beachten Sie, dass einzelne Bestandteile der Publikation anderweitigen Lizenz- bzw. urheberrechtlichen Bedingungen unterliegen können.

Zitieren

Zitierform:
Zitierform konnte nicht geladen werden.